Key Engineering Materials Online: 2009-10-08 ISSN: 1662-9795, Vols. 419-420, pp 557-560 doi:10.4028/www.scientific.net/kem.419-420.557 2010 Trans Tech Publications, Switzerland Utilizing Restricted Direction Strategy and Binary Heap Technology to Optimize Dijkstra Algorithm in WebGIS Li Rui Shenzhen Institute of Information Technology, Shenzhen 518029, China lir@sziit.com.cn Keywords: WebGIS; restricted direction strategy; shortest path; binary heap; Dijkstra algorithm Abstract. Shortest path is the core issue in application of WebGIS. Improving the efficiency of the algorithm is an urgent requirement to be resolved at present. By the lossy algorithm analyzing, which is the current research focus of the shortest path algorithm to optimize, utilizing adjacency table of storage structures, restricted direction strategy and binary heap technology to optimize the algorithm, thereby reduce the scale of algorithm to improve the operating efficiency of algorithm. This scheme has been applied in the simulation of the data downloaded from the Guangdong Provincial Highway Network Information System and satisfactory results have been obtained. Introduction With the popularization of Internet, Web Geography Information System (WebGIS) is being widely applied day by day due to its powerful function. As one of the most main functions of WebGIS, network analysis plays an important role on such aspects as traffic & transportation, program on urban pipeline network and management on land resource. The most basic issue in network analysis is the Shortest Path. As the basis for selecting the optimum in many fields, the Shortest Path has an important status in traffic network analysis system. The research of it has important theoretical value and application value. At present, there are many researches involving in the algorithms of searching the Shortest Path based on WebGIS. Among these researches, the Dijkstra s algorithm is one of the algorithms that are most suitable for searching the Shortest Path between two nodes in topological network. On the basis of the algorithm thought, people deduced several dozens of different optimized algorithms. The algorithms with excellent effect include DKA (the Dijkstra s algorithm implemented with approximate buckets), TQQ (graph growth with two queues), DKD (the Dijkstra s algorithm implemented with double buckets) [1-3], etc. The optimized algorithm on the Shortest Path raised in the paper is a hotspot of research at present, network feature is restricted by restricted direction search strategy. This algorithm lays emphasis on increasing time efficiency mostly. On the basis of the adjacency list storage structure, restricted direction strategy and binary heap technology are applied to optimize Dijkstra algorithm, in addition, the optimized algorithm is simulated by test, the two indices including memory and speed all attain satisfactory effects. Optimized path of Dijkstra algorithm Optimizing storage structure by applying adjacency list. For the data in road network, if the shortest path needs to be calculated, these data must be first abstracted into some data structure as per the relation between node and side, this is called establishing the topological relation of network in GIS. Because the network topology relation is generally abstracted into graph in the field of mathematics and computer, its basis is the storage representation of graph. The comparison of several storage ways on the merits and demerits are shown in Table 1 [4]. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (ID: 130.203.136.75, Pennsylvania State University, University Park, USA-06/03/16,07:41:17)
558 Advanced Design and Manufacture II Name matrix list Orthogonal list multilist Table 1 Comparison of several storage ways Realization method Merits Demerits Easily judging the relation Two-dimens Occupying large between both, easily ional array space finding vertex degree Saving space, easily Uneasily judging Linked list attaining the out-degree of the relation vertex between two points Linked list Small required space, With complex easily finding the in & out structure degree of vertex Linked list Saving space, easily With complex judging the relation structure between two points Time complexity O(n 2 + m n) or (m+n) or O( m+n) or O( m+n) According to the comparison on the above several storage structures, the algorithm applies the storage structure of adjacency list. By applying the structure, space may be saved, the out-degree of vertex may be attained, in addition, much time used for structural processing may be saved. Optimizing search scope by applying restricted direction strategy. According to Dijkstra algorithm and the characters of the road network itself, on the basis of analyzing the merits and localization of the Shortest Path algorithm in Refs. [5], [6] and [7], this paper uses restricted direction search algorithm. The core thought of the algorithm: Under the condition that the researched network may be approximately regarded as a plane network, taking the sum of the shortest path between the temporarily marked node and the source point plus the straight line distance between the temporarily marked node and the target node as one attribute value of the temporarily marked node, and this attribute value will be taken as the basis for selecting permanently marked node from the set of temporarily marked nodes, i.e. the method for selecting the temporary node with the smallest attribute value as the permanently marked node. As shown in Fig.1, R 1 and R n are temporary nodes, S 1, S n, C 1 and C n are respectively the straight line distances from the corresponding temporary nodes to source point and terminus, the linking line between R 1 and R n is D. Judgment principle of the original Dijkstra algorithm: if S 1 >S n, R n will be taken as the permanent marking node, subsequently, R n is taken as the source point for the next round of search; inversely, R 1 will be taken as the permanent marking node, and then R 1 is taken as the source point for the next round of search. Judgment method of Dijkstra search algorithm with restricted direction: if S 1 +C 1 >S n +C n, R n will be taken as the permanent marking node; if S 1 +C 1 <S n +C n, R 1 will be taken as the permanent marking node. Fig.1 The Dijkstra algorithm of restricted direction optimized strategy By comparison, it may be seen that the search scope in Dijkstra search algorithm with restricted direction is smaller than that in the original Dijkstra algorithm, because the search process of the original Dijkstra algorithm is similar to a series of concentric circles with source point as circle center; the Dijkstra search process with restricted direction is similar to a series of concentric circle
Key Engineering Materials Vols. 419-420 559 with source and terminus as focus. According to the principle of algorithm with restricted direction, its search process apparently tends to the terminus. Therefore, the efficiency of algorithm has been apparently increased. Optimizing priority queue by applying binary heap technology. It may be seen from the above algorithm that the core step in the process of realizing Dijkstra algorithm with restricted direction is to select a temporary node with the smallest attribute value. This is a process of cyclic comparison. In order to solve the problem, the most effective method is to conduct stack sequencing for temporary marking nodes, take the shortest path of node as the keyword of stack sequencing, so to constitute priority queue with incremental side weighted value, thus the point that meets requirements may be attained in each cycle, the performing efficiency of algorithm may be greatly increased. In this paper, binary heap technology is applied for the realization of priority queue. The method includes the following four operations:1sift(h):adjusting orderless sequence h into binary heap. 2Insert(x,h): inserting the element x into the stack h. 3Min(h): taking the element with the smallest keyword from stack, calling Sift(h) and adjusting it as a binary heap. 4Decrease(h,x,value): replacing the value of element x in the stack with smaller value. Realization of optimized algorithm According to the above thought, the realization of optimizing Dijkstra algorithm is shown as follows: Supposed that the starting point is S, the terminus is E and the binary heap is OPEN, all nodes are divided into three kinds of states, i.e. unmarked node(on), temporary node(fn) and marked node(in). Each node has several symbols: D(J), D 1 (J), D 2 (J), P(J), in which, D(J) is an approximate estimate of the optimum path P from S to E and via node J; D 1 (J) is the shortest path from S to J; D 2 (J) is an enlightenment function, i.e. the straight line distance from J to E; P(J) is the tending pointer of J, it points to the previous point of J in the shortest path from S to J. The concrete calculation step of the algorithm for finding the optimum path from S to E is shown as follows: STEP1: Initializing. The starting point S is set as follows: 1 D(S)=0, D 1 (S)=0, D 2 (S)=d se, P(S) is hollow; 2 Marking the starting point S, taking K=S, as IN state; 3 The initial binary heap OPEN is hollow. 4 All other nodes are set to ON. STEP2: Judging the states of all adjacent nodes J of K. If the state of J is ON, then D 1 (J)=D 1 (K)+d kj, D(J)=D 1 (J)+D 2 (J), P(J)=K. Marking the state of node J as FN, performing operation Insert(J,OPEN), where d kj is the straight line linking the distance from K to J; if the state of J is FN and D 1 (J)>D 1 (K)+d kj, D 1 (J)=D 1 (K)+d kj, then D(J)=D 1 (J)+D 2 (J), decrease(open,j,d 1 (K)+d kj ). For the tending pointer of node J, P(J)=K. STEP3: Selecting the next point. Taking node I with the smallest D(J) value in the stack, I =Min (OPEN), the point I is selected as one point in the shortest path and set to IN state. STEP4: If I=E, then shifting to STEP5, traversing the pointer P from E, until the starting point S, then the shortest path may be attained; or else, taking K=I, shirting to STEP2 for continuing. STEP5: End of algorithm. Test simulation analysis and result Test simulation. Recurring to the traffic road network among the main cities in Guangdong, this algorithm and the traditional Dijkstra algorithm were simulated. Different road data were put in the
560 Advanced Design and Manufacture II same computer to simulate and find the distance between the same starting point and terminus for many times. Table 2 shows the contrast between the operation time of two algorithm. Table 2 Running result of two algorithms Road mark Quantity of Quantity of arc Present Dijkstra(S) nodes sections algorithm(s) SMALRODE1 518 1624 1.24 1.09 MIDRODE2 945 2913 2.86 1.17 LONRODE3 1814 4603 3.52 1.36 Test conclusion. It may be seen from the data in Table 2 that two algorithms have similar operation time and the efficiency difference is not apparent when the quantity of nodes in the network is little. However, with the increase of quantity of nodes in the network, in theory, Dijkstra algorithm will traverse all nodes in the network, therefore, the efficiency will decrease quickly, while for the algorithm raised in the paper, due to the addition of restricted direction strategy and the optimized information of binary heap technology, the search direction will extend toward target node. In the road network figure with a lot of nodes, the quantity of search nodes may be reduced and the efficiency of search may be increased. Conclusions The algorithm on the optimization of the shortest path discussed in this paper has such merits as little memory occupation of storage structure, simple realization of algorithm and large speed, it may be effectively applied for searching the shortest path of road network structure which approximates to a plane network, it is a breakthrough in the application of the algorithm on the shortest path in WebGIS field. Of course, the realization on the function of searching the shortest path in GIS still needs further optimization according to the concrete features of different application fields. On the basis of unifying time complexity, the operational efficiency of algorithm shall be increased as far as possible, so to attain the search function module with the best comprehensive performance, satisfy the performance of system at maximum degree. This will have important actual sense. References [1] Chen Hong-ying, Yang Yi-min and Mao Ge-fei: Computer Engineering and Application Vol. 3 (2005),p. 208-211 [2] Zhan F B: Journal of Geographic Information and Decision Analysis Vol. 1 (1997), p. 69-82 [3] Yan Wei-min, Wu Wei-min: Data Structure(Tsinghua University Press, Beijing 2001). [4] Si Lian-fa, Wang Wen-jing: Bulletin of Surveying and Mapping Vol. 8 (2005), p. 15-18 [5] Li Cong-xin, Meng Xiang-gang and Zhang Ying: Journal of Shaanxi Institute of Technology Vol. 2 (2005), p. 18-20 [6] Zhu Jing: Computer and Modernization Vol. 9 (2005), p. 19-24 [7] Hu Shu-wei, Zhang Xiu-ru and Zhao Yang: Computer Technology and Development Vol. 16 (2006), p. 49-54
Advanced Design and Manufacture II 10.4028/www.scientific.net/KEM.419-420 Utilizing Restricted Direction Strategy and Binary Heap Technology to Optimize Dijkstra Algorithm in WebGIS 10.4028/www.scientific.net/KEM.419-420.557